226 Methods for the Solution of Special Problems [OH. vm 



Thus the lines of force inside the dielectric are all parallel to those of 



the original field, but the intensity is diminished in the ratio -^ - . The 



./L -\- ' 



field is shewn in fig. 78. 



IV. Nearly spherical surfaces. 



264. If r = a, the surface r = a + %, where % is a function of 6 and </>, will 

 represent a surface which is nearly spherical if % is small. In this case % 

 may be regarded as a function of position on the surface of the sphere r a, 

 and expanded in a series of rational integral harmonics in the form 



in which $ 1} $ 2 , ... are all small. 



The volume enclosed by this surface is 



_ 

 o 



If $ = 0, the volume is that of the original sphere r = a. 



The following special cases are of importance : 



r = a + ePj. To obtain the form of this surface, we pass a distance e cos 6 

 along the radius at each point of the sphere r = a. It is easily seen that 

 when e is small the locus of the points so obtained is a sphere of radius 

 a, of which the centre is at a distance e from the origin. 



r = a + a 1 S 1 . The most general form for a^ is Ix + my + nz, and this 

 may be expressed as aecos#, where 6 is now measured from the line of 

 which the direction cosines are in the ratio I : m : n. Thus the surface is the 

 same as before. 



r = a + S 2 . Since r is nearly equal to a, this may be written 



or # 2 + y 2 - + z 1 a 2 + an expression of the second degree. 



